2. Constrained optimization problems in smart grid

A lot of OEM problems in power grids can be represented by the following general form of constrained convex programming:

$$\min \sum\_{t=1}^{N} \sum\_{i=1}^{M} f\_i(\mathbf{x}\_i(t)) \tag{1}$$

$$\text{s.t.} \sum\_{i=1}^{M} \mu\_i \mathbf{x}\_i(t) = \xi(t) \tag{2}$$

$$
\omega\_i(t) \in \Omega\_i(t) \tag{3}
$$

where fi : R ! R are scalar, continuous, and convex functions, xi ∈ R are variables, t is the time index, ξð Þt is a known time-varying parameter, μ<sup>i</sup> . 0 are known parameters, Ωið Þt ⊂ R are local convex constraint sets which might be dependent on the previous time slot <sup>t</sup> � 1, x tð Þ≜½ � <sup>x</sup>1ð Þ<sup>t</sup> ; …; xMð Þ<sup>t</sup> T, and <sup>N</sup> and <sup>M</sup> are the number of time slots and variables.

Due to the global constraint (2), (1) is a centralized optimization problem. However, (1) is decomposable into subproblems corresponding to individual time slots; hence we can derive a sub-optimal solution to (1) by resolving it sequentially, i.e., from t ¼ 1 to t ¼ N. Doing so saves much time and computational costs in comparison with solving (1) for all time slots at once. Therefore, a sequential approach to solve (1) will be proposed in this chapter and will be applied to two problems, namely, DSWM and PSwEV. The former aims to maximize the benefits of both generators and consumers in transmission grids without explicitly considering ESSs, while the latter investigates the power scheduling in microgrids with EVs serving as mobile and distributed ESSs.

complex than that in traditional power grids. This urges the development of new concepts and approaches to deal with challenges that arose in smart grids. Some examples include demand side management (DSM) and real-time (dynamic) pricing (RTP) [2, 3]. As one of the main DSM activities [3–5], demand response (DR) motivates changes in electric use by end-use customers, in response to changes in the price of electricity over time, or gives incentive payments designed to induce lower electricity use at times of high market prices or when grid reliability is jeopardized [6]. DR not only shifts the load in peak times but also increases the

Traditionally, the OEM problem in transmission power networks is treated as the economic dispatch (ED) problem (see, e.g., [7]). Nonetheless, DSM including DR is often not included in the ED problem. To overcome this drawback, the social welfare maximization (SWM) problem is set up and solved, where the benefits from both the suppliers and the consumers are taken into account [8–14]. The effectiveness of the SWM problem has been verified in [15]. On the other hand, in distribution power networks, the OEM problem is usually investigated with the existence of an intermediate player called the aggregator who manages the power exchange between a small distribution grid and a larger or main grid. Therefore, the OEM problem in such situations means to maximize the benefits of the DER

To suppress the redundancy or to supplement the lack of energy from the fluctuating outputs of RESs, and to make the energy scheduling more flexible, ESSs are being introduced into smart grids as a solution. For instance, in power distribution systems or microgrids, EV's batteries are sources of mobile and distributed ESSs that might contribute to the energy efficiency, security, and flexibility of such distribution systems or microgrids. The stability and robustness of vehicle-to-grid (V2G) energy networks have been investigated in [16]. The problem of power scheduling with electric vehicle (PSwEV) in a microgrid has been introduced under a multi-agent setting in [17]. Due to the increase of market share of EVs and plug-in hybrid electric vehicles (PHEVs) predicted until 2050 [18], it is reasonable to believe that the PSwEV problem will be popular and important in the near future. To solve optimization problems arising in OEM problems, the existing methods can be classified depending on the centralized or distributed nature of such methods and the heuristic or analytical characteristic of the obtained solution. The security and resiliency of centralized approaches are weak due to a single point of failure and the huge communication and data processing, at the central unit. Those limitations can be improved by distributed approaches using multi-agent system (MAS), where

each agent corresponds to a bus or a portion of the grid and each agent only communicates with a few other neighboring agents. On the other hand, the heuristic methods (e.g., [19, 20]) usually require a long running time and local solutions, which are less effective than analytical methods for convex optimization problems. Thus, distributed and analytical approaches will be developed in the current chapter to solve convex optimization problems representing OEM problems in smart grids. A MAS-based distributed method for solving the ED problem in smart grid was proposed in [7] where the power losses are approximated by quadratic functions and the nonlinear coupling of oscillating agents is employed for decentralized solution derivation from the Karush-Kuhn-Tucker (KKT) conditions. The projected gradient methods were utilized in [10] to solve the SWM problem, where a MAS was utilized to derive the supply-demand mismatch in a distributed fashion. Another method named dual decomposition was used in [8] to get a distributed solution when the power balance is not strictly required. If the power balance is required, a distributed observer design was employed in [11] to estimate the power

grid's energy efficiency and customers' benefits.

Research Trends and Challenges in Smart Grids

owners, consumers, and the aggregator.

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